Consider an electric field $\vec{E} = 3 \times 10^3 \hat{i} \text{ N/C}$. What is the flux through a square of side $10 \text{ cm}$ in $Nm^2/C$ if the normal to its plane makes a $60^\circ$ angle with the $X$-axis?

State Gauss's law in electrostatics and provide its mathematical expression.

$A$ large charged plane having surface charge density $4.9 \times 10^{-6} \text{ C m}^{-2}$ lies in the $x-y$ plane. $A$ circular plane of radius $1 \text{ cm}$ is lying completely in the region where $x, y$ and $z$ coordinates are all positive. When the plane's normal makes an angle $60^{\circ}$ with the $z$-axis,the electric flux through the circular plane is (Given: $\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$)

The given figure shows the electric field lines due to two charges $q_1$ and $q_2$. What are the signs of the two charges?

$A$ point charge $+q$ is placed in front of a grounded metal plate. Points $P$ and $Q$ are between the charge $X$ and the plate $Y$ as shown in the figure. Let $E_P$ and $E_Q$ be the magnitudes of the electric fields at points $P$ and $Q$ respectively. Which of the following statements is true?

Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is $8.0 \times 10^{3} \; N m^{2} / C$.
$(a)$ What is the net charge inside the box?
$(b)$ If the net outward flux through the surface of the box were zero,could you conclude that there were no charges inside the box? Why or why not?